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Coexistence of near-field and far-field sources: the
angular resolution limit
Remy Boyer, Mohammed Nabil El Korso, Alexandre Renaux, Sylvie Marcos
To cite this version:
Remy Boyer, Mohammed Nabil El Korso, Alexandre Renaux, Sylvie Marcos. Coexistence of near-field and far-field sources: the angular resolution limit. International Workshop on New Computational Methods for Inverse Problem, 2013, Cachan, France. �hal-00926284�
Coexistence of Near-Field and Far-Field Sources: the
Angular Resolution Limit
R´emy Boyer\, Mohammed Nabil EL Korso], Alexandre Renaux\ and Sylvie Marcos\
\Laboratoire des Signaux et Systemes (L2S) ]Laboratoire SATIE, CNRS Universit´e Paris-Sud XI (UPS), CNRS, SUPELEC, Ecole normale suprieure de Cachan´
Gif-Sur-Yvette, France Cachan, France
E-mail: remy.boyer@lss.supelec.fr
Abstract. Passive source localization is a well known inverse problem in which we convert the observed measurements into information about the direction of arrivals. In this paper we focus on the optimal resolution of such problem. More precisely, we propose in this contribution to derive and analyze the Angular Resolution Limit (ARL) for the scenario of mixed Near- Field (NF) and Far-Field (FF) Sources. This scenario is relevant to some realistic situations.
We base our analysis on the Smith’s equation which involves the Cram´er-Rao Bound (CRB).
This equation provides the theoretical ARL which is independent of a specific estimator. Our methodology is the following: first, we derive a closed-form expression of the CRB for the considered problem. Using these expressions, we can rewrite the Smith’s equation as a 4-th order polynomial by assuming a small separation of the sources. Finally, we derive in closed-form the analytic ARL under or not the assumption of low noise variance. The obtained expression is compact and can provide useful qualitative informations on the behavior of the ARL.
1. Introduction
Very few works are related to the study of the realistic situation where there exists coexisting far- field (FF) and near-field (NF) sources [2] such as speaker localization using microphone arrays and guidance (homing) systems. At the contrary, we can find a plethora of contributions on the localization of far-field sources [1]. More recently, the problem of localization of near-field sources has been tackled in reference [4, 5] for instance. In the context of the problem of source localization, one can see three contributions: (1) propose new efficient algorithms/estimators [1], (2) study the estimation performance independently of a specific algorithm thanks to the lower bound on the Mean Square error (MSE) [3, 5] and (3) derive and study the theoretical resolution, i.e., the minimal angular distance to resolve/discriminate two closely spaced emitted signals in terms of their direction of arrivals. Our contribution belongs to the third point. More precisely, based on the Smith’s equation [6, 8, 9] which involves the Cram´er-Rao Bound (CRB) [3], we derive and analyze the Angular Resolution Limit (ARL) for the realistic scenario where we have two sources, one located in the far-field of the array and another in the near-field of the array.
2. Model setup
We consider some practical applications where the signals s1(t) and s2(t) with t ∈ [1 : T ] in which T is the number of snapshots received by an uniform linear array composed by L sensors are the mixture of near-field and far-field sources. More precisely, the (T L) × 1 observation vector is defined as y = ¯y + e where e is the complex centered circular additive white Gaussian noise of variance σ2 and the (T L) × 1 noise-free signal is ¯y =
As(1)
... As(T )
with s(t) = [s1(t) s2(t)]T and A = a(ω1) b(ω2, φ) where the signal sources sm(t) are viewed as deterministic known signals and the steering vectors are defined by [a(ω1)]` = eiω1` and [b(ω2, φ)]` = ei(ω2`+φ`2) with
` ∈ [0 : L − 1]. In addition, we assume ω1 6= ω2. We define the determinist non-zero separation by δ = ω2 − ω1. We focus our attention on the electric parameters but these parameters can be linked to the physical parameters, namely the DOA θm and the range r, according to ωm = −2πd/λ sin(θm) and φ = πd2/(λr) cos(θ2)2 where d is the distance inter-sensor and λ is the wavelength.
3. Analytic ARL in the Smith’s sense
3.1. Analytical expression of the Cramer-Rao Bound
The Cramer-Rao Bound (CRB) verifies the covariance inequality principle [3]. This bound is largely used in the signal processing community since it gives the best performance in term of Mean Square Error (MSE) at high Signal to Noise Ratio (SNR). Let ˆωm be an unbiased estimate of ωm, then
E{(ωm− ˆωm)2} ≥ [C]mm def= [J−1]mm (1) where E{.} is the mathematical expectation and J is the Fisher Information Matrix (FIM) defined by the Slepian-Bangs formula for a complex circular Gaussian observation y ∼ CN (¯y, σ2I):
[J]ij = 2 σ2<
( ∂¯y
∂ωi
H
∂¯y
∂ωj
)
= 2 σ2
T
X
t=1
<
(
s(t)H ∂A
∂ωi
H
∂A
∂ωj
s(t) )
(2)
where ∂ω∂A
1 = ˙a(ω1) 0, ∂ω∂A
2 =0 ˙b(ω2, φ), ∂A∂φ =0 ¨b(ω2, φ) where ˙a(ω1) (resp. ˙b(ω2, φ)) is the first-order derivative of a(ω1) (resp. b(ω2, φ)) w.r.t. the parameters ω1, ω2, respectively and ¨b(ω2, φ) is the first-order derivative w.r.t. the parameter φ. After some algebra, one obtains the CRB for the parameters of interest and the coupling term:
CRB(ω1) = σ2 2
β
Q, (3)
CRB(ω2) = σ2 2
L2||s1||2−Lη2(δ)
3||s2||2
Q , (4)
CRB(ω1, ω2) = −σ2 2
ζ(δ) − η(δ)LL 3
4
Q . (5)
where β = ||s2||2
L2−LL23
4
, Q = β
L2||s1||2−Lη2(δ)
3||s2||2
−
ζ(δ) − η(δ)LL2
3
2
, Lr = PL−1
`=0 `r and η(δ) = <n
hPL−1
`=0 `2ei(δ`+φ`2)o
and η(δ) = <n
hPL−1
`=0 `3ei(δ`+φ`2)o
with h = sH1 s2.
3.2. Derivation of the analytic ARL
3.2.1. The Smith’s equation: The Smith’s methodology [6] provides the ARL as the solution of the following equation:
CRB(δ) = δ2. (6)
Expanding CRB(δ) w.r.t. the localization parameters and the coupling term, we have [7]:
CRB(δ) = CRB(ω1) + CRB(ω2) − 2CRB(ω1, ω2) (7) where the CRBs are given by expressions (3)-(5).
3.2.2. Polynomial resolution of the linearized problem: Solving analytically equation (6) with (7) and (3)-(5) seems an intractable problem. Recalling that we assume that the separation δ is small then a first-order Taylor expansions of ζ(δ) and η(δ) lead to the following approximations:
η(δ) ≈ < {h(v(φ) + jδr(φ))} , ζ(δ) ≈ < {h(u(φ) + jδv(φ))} (8) in which u(φ) =PL−1
`=0 `2eiφ`2, v(φ) =PL−1
`=0 `3eiφ`2and r(φ) =PL−1
`=0 `4eiφ`2. The above relation can be rewritten in a linear form w.r.t. the separation according to
< {h(v(φ) + jδr(φ))} = P − δQ, < {h(u(φ) + jδv(φ))} = P0− δQ0 (9) where P = <{hv(φ)}, Q = −={hr(φ)}, P0 = <{hu(φ)} and Q0 = −={hv(φ)}. Using these approximations, one can rewrite the CRB as
C = J−1 def= σ2 2
Q−1 ×
× ×
, (10)
where, the Schur complement can be approximated as follow Q ≈ P2(δ) P1(δ) P1(δ) β
in which we have introduced the two following polynomials P2(δ) = α1δ + α0 and P2(δ) = a2δ2+ a1δ + a0 where a2 = −L Q2
4||s2||2, a1 = −L2P Q
4||s2||2, a0 = L2||s1||2−L P2
4||s2||2, α1 = Q0−LL3
4Q and α0 = P0−LL3
4P . The linearized CRB expressions are now given by
CRB(ω1) ≈ σ2 2
β
Q(δ), (11)
CRB(ω2) ≈ σ2 2
P2(δ)
Q(δ), (12)
CRB(ω1, ω2) ≈ −σ2 2
P1(δ)
Q(δ), (13)
where Q(δ) = P2(α)β −P12(δ) = (βa2−α21)δ2+(βa1−2α0α1)δ +βa0−α20. Consequently from (6) and (11)-(13), the Smith’s equation becomes δ2= QQ(δ)0(δ) where Q0(δ) = σ22(β + P2(δ) + 2P1(δ))def= c2δ2 + c1δ + c0 with c2 = σ22a2, c1 = σ22(a1+ 2α1) and c0 = σ22(β + a0+ 2α0). So, it is easy to see that the Smith’s equation provides the ARL as the solution of a 4-th order polynomial according to
R(x) = Q(x)x2− Q0(x)def= x4+ g3x3+ g2x2+ g1x + g0 (14) where x is a free variable with g0 = −βac0
2−α21, g1 = −βac1
2−α21, g2 = βaβa0−α20−c2
2−α21 , g3 = βaβa1−2α0α1
2−α21
where we assume βa2 6= α21.
3.2.3. Analytic solutions of R(x): The resolution of a 4-th order polynomial is not straightforward but we propose a solution to this problem. More precisely, as CRB(δ) = CRB(−δ), it is easy to see that if δ is a solution of the Smith’s equation then −δ is also a solution. This implies that if δ is a root of R(x) then −δ is also a root. So, R(x) has four roots, namely {δ, −δ, r1, r2} and a decomposition of this polynomial into a product of monomial terms is given by
R(x) = (x − δ)(x + δ)(x − r1)(x − r2) (15)
= x4− (r1+ r2)x3+ (r1r2− δ2)x2 (16) + δ2(r1+ r2)x − r1r2δ2. (17) We can identify the coefficients of the polynomials (14) and (17) according to
−(r1+ r2) = βaβa1−2α0α1
2−α21 , r1r2− δ2 = βaβa0−α20−c2
2−α21 , δ2(r1+ r2) = − c1
βa2−α21,
−r1r2δ2 = − c0
βa2−α21.
(18)
Combining the second and the last equations, we have to solve the following polynomial:
R0(x) = (βa2− α21)x4+ (βa0− α20− c2)x2− c0. (19) Polynomial R0(x) can be reformulated as a 2-rd order polynomial according to
R0(z) = (βa2− α21)z2+ (βa0− α20− c2)z − c0. (20) The discriminant is
∆ = (βa0− α20− c2)2
1 + 4 (βa2− α21)c0
(βa0− α20− c2)2
. (21)
The study of the sign of the discriminant is not straightforward but observe that c0, c2 σ2→0
−→ 0, so if the noise variance is low then the discriminant is given by
∆σ−→ (βa2→0 0− α20− c2)2≥ 0. (22) So, we know that if the noise variance is not too high, there exists two candidates for the ARL (or a double solution if the discriminant is zero) which are given by
δ = s
−(βa0− α20− c2) ±√
∆
2(βa2− α21) (23)
where √
∆ is given by the square root of expression (21).
3.2.4. Analytic expression of the ARL: To discriminate the two possible solutions, we advocate that δ σ
2→0
−→ 0 is a reasonable property. Thus, note thanks to (22), the sign in (23) must be chosen as positive to ensure δ σ−→ 0. If not (in case of negative sign in (23)) the chosen solution will2→0 be meaningless. So, finally the ARL is given by
δ = s
−(βa0− α20− c2) +√
∆
2(βa2− α21) . (24)
To provide further simplifications, we can see that
√
∆ ≈ βa0− α02− c2+ 2(βa2− α12)c0
βa0− α20− c2 (25)
thanks to a first-order Taylor expansion of the square root√
1 + x ≈ 1 + 1/2x for small x. Using the above approximation of √
∆, the ARL takes the simple following expression:
δ ≈
r c0
βa0− α20− c2. (26)
So, we can see that δ ≈ O(σ).
4. Numerical illustrations
In this simulation part, we have considered an array constituted by L = 10 sensors. The sources are chosen to be close where θF F = π/3 and θN F = π/3.1 and the range of this source belongs to the interval [0.62(d3(L − 1)3/T )λ, 2d2(L − 1)2/λ] [5] where d = 0.0125 m and the carrier frequency is f0 = 10 Mhz (the wavelength is λ = c/f0 where c is celerity of the light). The modulus of the NF sources is fixed to ten times higher than the modulus of FF source. It is normal to assume that the source which is the closest has a higher power than the one which belongs to the FF. The number of snapshot is T = 100. On Fig. 1, we have reported the positive roots of each polynomials R(x), R0(x) and the analytic ARL given in expression (24). As we can see one root for polynomials R(x) and R0(x) are independent from the noise variance. These two roots have to be ignored. In addition, we can see that the one root for R(x) and R0(x) are equal and follow a decreasing function w.r.t. the inverse of the noise variance. Moreover, the analytic ARL given in expression (24) is in good agreement with these roots and thus assess the validity of the derivations given in the previous section. The ARL given in expression (24) and the approximated ARL under the assumption of low noise variance given in expression (26) are reported on Fig. 2. It is important to highlight the good accuracy of the proposed approximated ARL.
5 10 15 20 25 30 35 40
10−3 10−2 10−1 100 101
1/σ2
Positive roots
R(x) R’(x)
ARL : δ
5 10 15 20 25 30 35 40
0.005 0.01 0.015 0.02 0.025
1/σ2 ARL
ARL (low noise variance)
Figure 1. (a) Positive roots for polynomials R(x), R0(x) and analytic ARL given in expression (24) vs. the inverse of the noise variance. (b) ARL given in expression (24) and the approximated ARL under the assumption of low noise variance given in expression (26) vs. the inverse of the noise variance.
5. Conclusion
In this paper, we have derived and analyzed the Angular Resolution Limit (ARL) based on the resolution of the linearized Smith’s equation for the new and realistic scenario where far-field and near-field sources are mixed. Generally speaking, the ARL is important and fundamental since this quantity gives the limit in the resolvability/separation of two closely spaced signals in term of their direction of arrivals. We show that for the chosen application, the resolution of the Smith’s equation turns to be the selection of the ”right” root of a 4-th order polynomial.
This allows us to give a closed-form (analytic) expression of the ARL.
6. References
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[3] P. Stoica and R.L. Moses, Spectral Analysis of Signals, Prentice Hall, 2005.
[4] Y. D. Huang and M. Barkat, ”Near-field multiple source localization by passive sensor array”, IEEE Trans.
Antennas Propag., vol. 39, no. 7, pp. 968-975, Jul. 1991.
[5] E. Grosicki, K. Abed-Meraim, and Y. Hua, ”A weighted linear prediction method for near-field source localization”, IEEE Trans. Signal Process., vol. 53, pp. 3651- 3660, Oct. 2005.
[6] S. T. Smith, ”Statistical resolution limits and the complexified Cramer-Rao bound”, IEEE Trans. Signal Process., vol. 53, pp. 1597-1609, May 2005.
[7] EL Korso, M. N., Boyer R., Renaux A. and Marcos S. ”Statistical Resolution Limit for the Multidimensional Harmonic Retrieval Model: Hypothesis Test and Cramer-Rao Bound Approaches”, EURASIP Journal on Advances in Signal Processing, special issue ”Advances in Angle-of-Arrival and Multidimensional Signal Processing for Localization and Communications”, vol. 2011, no1, 2011, pp. 1-14
[8] R. Boyer, ”Performance bounds and angular resolution limit for the moving co-located MIMO radar”, IEEE Trans. Signal Process., vol. 59, no. 4, pp. 15391552, 2011.
[9] M. N. El Korso, R. Boyer, A. Renaux and S. Marcos, ”Statistical resolution limit of the uniform linear cocentered orthogonal loop and dipole array”, IEEE Trans. Signal Process., Volume: 59, Issue: 1, Jan.
2011, pp. 425-431.
